The Broadway transfer of the National Theatre's War Horse, the moving first world war story of a farm boy and his horse, where the puppet and human actors regularly leave their audience drenched in tears, has scooped best play – and five other accolades – at this year's Tony awards in New York.

There were also laurels for another very British production, Jerusalem, Jez Butterworth's anarchic Arcadian fantasy first seen at the Royal Court, which won best actor for its star, Mark Rylance. Rylance's extraordinary performance as the charming, dangerous, drug-dealing and tale-telling Rooster Byron has been the talk of Broadway – and his performance at the Tonys ceremony on Sunday night matched it.

He quoted a prose poem by the American poet Louis Jenkins in his acceptance speech – just as he did three years ago when he also took the best actor Tony, for Boeing-Boeing – explaining later that he thought things had got a bit boring.

"Unlike flying or astral projection, walking through walls is a totally earth-related craft, but a lot more interesting than pot-making or driftwood lamps," he recited, words taken from Jenkins's Walking Through A Wall. "The worst things are wire fences, maybe it's the molecular structure of the alloy or just the amount of give in a fence, I don't know, but I've torn my jacket and lost my hat in a lot of fences."

Dominic Cooke, artistic director of the Royal Court, which is co-producing the show on Broadway, said: "Mark thoroughly deserves the recognition he has received today. His Johnny 'Rooster' Byron is a tour de force and his performance remains with you long after the show is over."

War Horse, which has been a phenomenal success since it opened at the National Theatre in 2007, is based on the 1982 children's novel by Michael Morpurgo, about a farm boy from Devon who follows his beloved horse into the maelstrom of the first world war.

It took a total of six Tonys, including one for Handspring, the company that created the extraordinary puppets. The best director prize was shared by Marianne Elliott and Tom Morris, who at the customarily tear-sodden theatrical get-together, said: "We quite like it when people cry."

War Horse is still running in the West End, where it has sold more than 1m tickets, and is selling out on Broadway every night. A film adaptation by Steven Spielberg, using real horses, is due for release later this year.

The biggest winner of the night, with nine awards, was an improbable – and, to some, bitterly controversial – religious-themed musical, The Book of Mormon, created by Trey Parker and Matt Stone, the team behind the blackly humorous animation South Park. Parker thanked Joseph Smith, the founder of the Mormon church: "You did it, Joseph!" he cried, looking towards the heavens. There are plans for a London production next year.

Spider-Man: Turn Off the Dark, the most talked about musical on Broadway, for all the wrong reasons, featured in the awards albeit only in a performance by some of the cast. The show, based on the comic book character, and with music by Bono and the Edge of U2, was ineligible for nomination because, a year after its original planned opening, after innumerable rewrites, cast injuries, losing its original director, and more than six months of previews, it has yet to open officially.

• This article was amended on 14 June 2011. The original said that War Horse took a total of five Tonys, and that it has sold more than £1m worth of tickets. This has been corrected.